A complete and statistically consistent uncertainty quantification for deep learning is provided, including the sources of uncertainty arising from (1) the new input data, (2) the training and testing data (3) the weight vectors of the neural network, and (4) the neural network because it is not a perfect predictor. Using Bayes Theorem and conditional probability densities, we demonstrate how each uncertainty source can be systematically quantified. We also introduce a fast and practical way to incorporate and combine all sources of errors for the first time. For illustration, the new method is applied to quantify errors in cloud autoconversion rates, predicted from an artificial neural network that was trained by aircraft cloud probe measurements in the Azores and the stochastic collection equation formulated as a two-moment bin model. For this specific example, the output uncertainty arising from uncertainty in the training and testing data is dominant, followed by uncertainty in the input data, in the trained neural network, and uncertainty in the weights. We discuss the usefulness of the methodology for machine learning practice, and how, through inclusion of uncertainty in the training data, the new methodology is less sensitive to input data that falls outside of the training data set.

翻译：本文提出了一种完整且统计一致的不确定性量化方法，用于深度学习模型，涵盖以下不确定性来源：(1) 新输入数据，(2) 训练与测试数据，(3) 神经网络权重向量，以及(4) 神经网络本身作为非完美预测器所产生的误差。利用贝叶斯定理与条件概率密度，我们系统阐述了如何量化各项不确定性来源。同时，我们首次提出一种快速实用的方法，用于整合所有误差源。为作说明，新方法被应用于量化云自动转化率预测中的误差，该预测基于人工神经网络——该网络使用亚速尔群岛飞机云探针测量数据进行训练，并以随机收集方程构建的双矩分档模型为基础。在此具体案例中，训练与测试数据的不确定性对输出影响最大，其次为输入数据的不确定性、已训练神经网络的不确定性，以及权重的不确定性。我们讨论了该方法对机器学习实践的价值，并说明通过纳入训练数据的不确定性，新方法对超出训练数据集范围的输入数据具有较低的敏感性。